$|2^x-3^y|=1$ has only three natural pairs as solutions

Question

Consider the equation $$|2^x-3^y|=1$$ in the unknowns $x \in \mathbb{N}$ and $y \in \mathbb{N}$. Is it possible to prove that the only solutions are $(1,1)$, $(2,1)$ and $(3,2)$?

Answer 1

Yes. Levi ben Gerson (1288-1344), also known as Gersonides, proved this. The Gersonides proof can be found here.

EDIT: miracle notes that the link is broken. This link works (for now). But to keep everyone happy, I'll paste in the relevant parts of what's there:

In 1342, Levi ben Gerson, also known as Gersonides (1288-1344), proved that the original four pairs of harmonic numbers are the only ones that differ by 1. Here's how his proof went (using modern notation). [note --- "harmonic numbers" are those that can be written as $2^n3^m$]

If two harmonic numbers differ by 1, one must be odd and the other even. The only odd numbers are powers of 3. Hence one of the two harmonic numbers must be a power of 3 and the other a power of 2. The task then involves solving two equations:

$2^n = 3^m + 1$ and $2^n = 3^m - 1$.

Gersonides had the idea of looking at remainders after division of powers of 3 by 8 and powers of 2 by 8. For example, 27 divided by 8 gives a remainder of 3. For powers of 3, the remainders all turn out to be 1 or 3, depending on whether the power is even or odd. The remainders for powers of 2 are 1, 2, 4, then 0 for all powers higher than 2.

For $2^n = 3^m + 1$, when $m$ is odd, $3^m$ has remainder 3, and $2^n = 3^m + 1$ must then have the remainder 4, so $n = 2$ and $m = 1$. That gives the consecutive harmonic numbers 3, 4. When $m$ is even, the equation gives the consecutive harmonic numbers 1, 2.

For $2^n = 3^m - 1$, when $m$ is odd, $3^m$ has remainder 3, so $2^n = 3^m - 1$ has remainder 2; as a result, $n = 1$ and $m = 1$, to give the consecutive harmonic numbers 2, 3. The final case, when $m$ is even, is a little trickier and requires substituting $2k$ for $m$, then solving the equation $2^n = 3^{2k} - 1 = (3^k - 1)(3^k + 1)$. That gives the consecutive harmonic numbers 8, 9. QED.

Answer 2

This is an easy consequence of Catalan's conjecture, which was famously proved by Preda Mihăilescu in 2002 (so now it's also called Mihăilescu's theorem):

$3^2-2^3=1$ is the only solution to $x^a - y^b = 1$ for integers $a, b, x, y \ge 2$.

Answer 3

Assume that $x>3$ and $y>2$.

$3^2=1\pmod{8}$. Since $3\not=-1\pmod{8}$ we know that $3^y\not=-1\pmod{8}$. Thus, if $|2^x-3^y|=1$, we must have $2^x-3^y=-1$ and $y$ must be even. Thus, we get that $$ 2^x=(3^{y/2}-1)(3^{y/2}+1)\tag{1} $$ The only factors of a power of $2$ are other powers of $2$, and the only powers of two that differ by $2$ are $2$ and $4$, but since $y>2$, we have $3^{y/2}-1>2$ and $3^{y/2}+1>4$.

Therefore, there are no $x>3$ and $y>2$ so that $|2^x-3^y|=1$.

Answer 4

Case 1: If $2^x=3^y+1$ then $x \ge 1$. If $y=0$ then $x=1$. If $y \ge 1$ then $3^y+1 \equiv 1 \pmod{3}$. Therefore $2^x \equiv 1 \pmod{3}$. Hence $x=2x_1$ with $x_1 \in \mathbb{N}^*$. The equation is equivalent to $$3^y= (2^{x_1}-1)(2^{x_1}+1)$$ Since $2^{x_1}+1>2^{x_1}-1$ and $\gcd (2^{x_1}-1,2^{x_1}+1) \ne 3$ then $2= 3^{m} (3^{n-m}-1)$ with $2^{x_1}-1=3^m,2^{x_1}+1=3^n$. Thus, $m=0,n=1$. We obtain $x_1=1$ or $x=2$. Thus, $y=1$.

Case 2. If $3^y=2^x+1$. If $x=0$ then there is no natural number $y$. If $x=1$ then $y=1$. If $x \ge 2$ then $3^y \equiv 1 \pmod{4}$. Thus, $y$ is even, let $y=2y_1$ with $y_1 \in \mathbb{N}^*$. The equation is equivalent to $$2^y=(3^{y_1}-1)(3^{y_1}+1)$$ Thus, $2=2^n(2^{m-n}-1)$ with $2^m=3^{y_1}+1,2^n=3^{y_1}-1$. It follows that $n=1,m=2$. Thus, $y_1=1$ or $y=2$. We get $x=3$.

The answer is $\boxed{ (x,y)=(1,0),(1,1),(2,1),(3,2)}$.

Answer 5

Here is a solution using Zsigmondy's theorem. Recall that one form of the theorem tells us $a^n+b^n$ has a prime factor $p$ that does not divide any of $a^k+b^k$ for $k<n$, with the exception of$2^3+1^3=9$.

If $2^x=3^y+1$, then by the theorem for $y\geq 2$, there is a prime $p$ that divides $3^y+1$ and does not divide $3^1+1=4$, i.e. $p$ is odd. But only prime that divide $3^y+1=2^y$ is $2$, contradiction. We can see that the exception in the theorem does not apply here. So $y=1$, hence $2^x=4$, i.e. $x=2$.

If $3^y=2^x+1$, then first note that the exception in the theorem gives a solution $y=2$ and $x=3$. Otherwise, by similar reasoning as before for $x \geq 2$ we must have a prime $p$ dividing $2^x+1$ such that $p\neq 3$. This is impossible and so $x=1$, which gives $y=1$.

Thus only solutions for $(x,y)$ are $(2,1)$, $(3,2)$ and $(1,1)$.